{"title":"Colengths of fractional ideals and Tjurina number of a reducible plane curve","authors":"Abramo Hefez, Marcelo Escudeiro Hernandes","doi":"arxiv-2409.11153","DOIUrl":null,"url":null,"abstract":"In this work, we refine a formula for the Tjurina number of a reducible\nalgebroid plane curve defined over $\\mathbb C$ obtained in the more general\ncase of complete intersection curves in [1]. As a byproduct, we answer the\naffirmative to a conjecture proposed by A. Dimca in [7]. Our results are\nobtained by establishing more manageable formulas to compute the colengths of\nfractional ideals of the local ring associated with the algebroid (not\nnecessarily a complete intersection) curve with several branches. We then apply\nthese results to the Jacobian ideal of a plane curve over $\\mathbb C$ to get a\nnew formula for its Tjurina number and a proof of Dimca's conjecture. We end\nthe paper by establishing a connection between the module of K\\\"ahler\ndifferentials on the curve modulo its torsion, seen as a fractional ideal, and\nits Jacobian ideal, explaining the relation between the present approach and\nthat of [1].","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"187 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Algebraic Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.11153","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this work, we refine a formula for the Tjurina number of a reducible
algebroid plane curve defined over $\mathbb C$ obtained in the more general
case of complete intersection curves in [1]. As a byproduct, we answer the
affirmative to a conjecture proposed by A. Dimca in [7]. Our results are
obtained by establishing more manageable formulas to compute the colengths of
fractional ideals of the local ring associated with the algebroid (not
necessarily a complete intersection) curve with several branches. We then apply
these results to the Jacobian ideal of a plane curve over $\mathbb C$ to get a
new formula for its Tjurina number and a proof of Dimca's conjecture. We end
the paper by establishing a connection between the module of K\"ahler
differentials on the curve modulo its torsion, seen as a fractional ideal, and
its Jacobian ideal, explaining the relation between the present approach and
that of [1].